Dynamic Fracturing Simulation of Brittle Material using the Distinct Lattice Spring Method with a Full Rate-Dependent Cohesive Law

O R I G I N A L P A P E R

Dynamic Fracturing Simulation of Brittle Material using

the Distinct Lattice Spring Method with a Full Rate-Dependent

Cohesive Law

T. Kazerani•_{G. F. Zhao}•_{J. Zhao}

Received: 22 January 2010 / Accepted: 19 April 2010 / Published online: 7 May 2010  Springer-Verlag 2010

Abstract A full rate-dependent cohesive law is imple-mented in the distinct lattice spring method (DLSM) to investigate the dynamic fracturing behavior of brittle materials. Both the spring ultimate deformation and spring strength are dependent on the spring deformation rate. From the simulation results, it is found that the dynamic crack propagation velocity can be well predicted by the DLSM through the implemented full rate-dependent cohesive law. Furthermore, a numerical investigation on dynamic branching is also conducted by using the DLSM code. Keywords Dynamic fracturing  Crack propagation velocity Rate-dependent cohesive law  DLSM

1 Introduction

A major objective in studying dynamic fracturing is to predict crack dissemination pattern and velocity. Linear elastic fracture mechanics (LEFM) theoretically suggests the Rayleigh surface wave speed vR as the upper limit for the material crack velocity vcr (Freund 1990; Broberg

1999). However, in rapid cracking of brittle solids, the crack tip zone undergoes a series of complex mechanisms such as large and nonlinear deformation, high strain rate, plasticity, micro-crack nucleation, heat diffusion, and subsequent thermal softening. These mechanisms cause the fracture energy Gf to increase when the crack velocity

increases. For most materials, when vcr approaches a crit-ical value vL, Gfinfinitely increases, and finally, the crack stops further accelerating (Ravi-Chandar and Knauss1984; Dally et al. 1985; Fineberg et al.1991; Shioya and Zhou

1995; Sharon et al.1996).

Several approaches have been so far applied to examine dynamic fracturing. Nishioka (1995) and Nishioka et al. (2001) introduced the dynamic J-integral into a moving finite element mesh that needs a re-meshing algorithm with a very fine mesh around the crack tip. Belytschko and Black (1999) and Moes et al. (1999) suggested an extended FEM (XFEM) solution with crack propagation through ordinary elements which gives a smooth crack path with minimal re-meshing effort. However, discontinuities must be con-sidered in the shape functions to model broken elements. It makes the solution too effortful when cracks intersect.

In resent years, cohesive finite element methods (CFEM) which are based on the cohesive zone concept, invented by Dugdale (1960) and Barenblatt (1962), have been extensively used. In a cohesive element, crack sepa-ration is expressed by a cohesive law linking crack surface traction to its opening displacement. Xu and Needleman (1994, 1995, 1996) have applied an elastic-exponentially decaying cohesive law. Camacho and Ortiz (1996,1997) proposed a rigid-linear model law in 2D space. Later on, Pandolfi et al. (1999,2000) and Ruiz et al. (2000) carried that out in 3D analysis. Zhai et al. (2004) analyzed dynamic fracture in two-phase Al2O3/TiB2 ceramic composite microstructures using a CFEM. The model integrates cohesive surfaces along all finite element boundaries as an intrinsic part of the material description. This approach obviates the need for any specific fracture criteria and assigns models with the capability of predicting fracture paths and fracture patterns. Karedla and Reddy (2007) applied high inertia zone theory to a cohesive finite

T. Kazerani (&)  G. F. Zhao  J. Zhao Laboratoire de Me´canique des Roches (LMR), Ecole Polytechnique Fe´de´rale de Lausanne (EPFL), EPFL/ENAC/IIC/LMR, Station 18,

1015 Lausanne, Switzerland e-mail: tohid.kazerani@epfl.ch DOI 10.1007/s00603-010-0099-0

element model to examine crack branching in PMMA. However, crack propagation speed was not fully discussed. In all the research mentioned above, the cohesive laws used are rate-independent, i.e., the traction mobilized within the crack is just a function of its separation and independent of its opening speed. Much attention has been paid recently to the rate-dependent cohesive models. Li and Bazant (1997a,b) proposed a visco-elastic rate-dependent law to analyze a stationary crack under dynamic loading in concrete that results in cohesive forces extremely exceed-ing the material ultimate strength. Lee and Prakash (1999) studied dynamic fracturing in high-strength brittle steels. They found that the rate-dependency assumption was cru-cial to reproduce the experimental results. Zhou et al. (2005) introduced a linear decaying law into the cohesive interface elements to reproduce the laboratory results of PMMA cracking. They emphasized the necessity of con-sidering rate-dependency to restrict the crack velocity. However, numerically predicted vcrwas still larger than vL. Block et al. (2007) proposed the cohesive nodal force method, in which they assigned a continuum damage model to the solid elements, to control the model energy dissipation at high values of the loading rate.

In all the rate-dependent models reviewed, only the crack cohesive displacement (i.e., separation) is changing with the crack opening rate, and the crack cohesive strength is held fixed. This is contrary to most experimental observations which confirm that both the cohesive stress and displacement increase with the opening velocity (Li and Bazant 1997a, b; Costanzo and Walton 1998; Allen and Searcy 2001; Kubair et al. 2003; Xu et al. 2003; Ivankovic et al.2004).

The aim of this study is to develop a ‘‘full’’ rate-dependent law in which both the cohesive stress and the displacement of the contact depend on its opening rate (Kazerani and Zhao

2010). This model is to be introduced into the distinct lattice spring method (DLSM) (Zhao et al.2010a; Zhao and Zhao

2009) for simulating dynamic fracturing of the brittle mate-rials. In coming sections, the experimental work done by Shioya and Zhou (1995) on dynamic crack propagation in PMMA plates is briefly reviewed. Following this, the DLSM methodology and the proposed rate-dependent cohesive law are explained in detail. Finally, the developed numerical model is used to reproduce the laboratory data. The simula-tion results show that our model is capable of reproducing the actual fracture energy and crack propagation velocity observed in the experiments.

2 Dynamic Crack Propagation in PMMA Plates In this section, the experimental work done by Shioya and Zhou (1995) on fracturing behavior of pre-strained PMMA

strips is presented. More explanation in terms of the test setup and the experimental results is published by Zhou et al. (2005), and a detailed description is presented by Zhou (1996).

2.1 Experimental Setup and Measurement

Polymethyl methacrylate (PMMA) rectangular plate (3 mm thick) was used as the test material. Before crack propagation, a universal test machine loads the specimen in tension. The magnitude of the loading is measured by the load cell in the test machine. After reaching a given load level, a small sharp crack is initiated at the middle point of one specimen end by a razor. Because the material is very brittle, the small crack propagates straight across the specimen.

To record the crack propagation during the test, electric conductive lines are drawn on the specimen surface. They are connected to an electronic logic circuit. The output signal is recorded by a digital memory. The time when the crack cuts each line position is recorded in the digital memory. From this record, and the position of each line, the propagation history of the crack and its speed is obtained.

The data show that the crack velocity tends to a steady asymptotic value, and this steady propagation state con-tinues even when the crack tip reaches the opposite edge of the specimen. The average steady velocity of the crack is called crack velocity and denoted by vcr.

2.2 Test Results

Various specimens with different geometries are tested. The specimens are loaded to different levels of preloading before initiating the crack. By changing the tensile load-ing and the specimen shape, different values of elastic strain energy, W, are obtained. As the prescribed dis-placement is small compared to the specimen size, we consider the kinetic energy of the material in the region behind the crack front to be negligible. Therefore, the strain energy stored per unit length of the specimen is approximately equal to the energy consumed by the unit length of the crack propagation, i.e., the dynamic fracture toughness Gf. This equality is verified by Zhou et al. (2005) in detail.

From each test, a data point of (vcr, Gf) is obtained. As shown in Fig. 1, these points are fitted with a monotoni-cally-increasing curve which can be expressed by an empirical equation as follows:

Gf ¼ G0

vL

vL vcr

ð1Þ where vL= 675 m/s and G0= 1,000 N/m.

3 The Distinct Lattice Spring Method (DLSM)

The DLSM (Zhao et al.2010a; Zhao and Zhao2009) is a microstructure-based numerical model based on the real-istic multidimensional inter bond (RMIB) model (Zhao et al. 2010b). The RMIB model is an extension of the virtual multidimensional inter bond (VMIB) model (Zhang and Ge 2005) in which materials are discretized into mass particles (see Fig.2). Whenever two particles are detected in contact, they are linked together through bonds between their center points. The multi-dimensional internal bond is adopted which includes one normal bond and one shear bond for both 2D and 3D. Due to the explicit consideration of the material microstructure, the proposed micromechanical model has the potential to give a more realistic approximation of the material failure behavior than a phenomenological model. Based on Cauchy-born rules and hyper-elastic theory, the relation-ship between the micromechanical parameters and the macro material constants, i.e., the Young’s modulus and the Poisson ratio, can be obtained as follows (Zhang and Ge 2005): kn¼ 3E a3D_{ð1 2vÞ} and ks¼ 3 1ð  4vÞE a3D_{ð1þ vÞ 1  2vð Þ} ð2Þ

where knand ks are the normal and shear stiffness of the inter-particle bond, respectively. E is the material Young’s modulus, v the Poisson ratio and a3Dis the microstructure geometry coefficient which is obtained from

a3D_¼

P_l2 i

V ð3Þ

where li denotes the initial length (before loading) of the bonds, and V is the volume of the model.

The particles and springs represent the material. The equation of motion of the system of particles and springs is expressed as

½Ku þ ½C _u þ ½M€u¼ FðtÞ ð4Þ

where u represents the particle displacement vector, [M] the diagonal mass matrix, [C] the damping matrix, and F(t) the vector of all external forces acting on the particles.

In the DLSM, the motion equations of the particle sys-tem are solved through the explicit central finite difference scheme. The calculation cycle is illustrated in Fig.2. Given the particle displacements (either prescribed initially or obtained from the previous time step), new contacts and broken bonds are detected. The list of neighboring particles for each particle is updated. Then the contact and spring forces between the particles are calculated according to the prescribed force–displacement relations. The interaction between the particles is represented by one normal spring and one shear spring. The shear spring is a multi body spring which is different from the conventional lattice spring methods. Multi body means that the shear dis-placement is calculated from the local strain state which is determined from the displacement of a cloud of particles. Multi body spring is necessary to keep the rotational invariance and to allow the model to handle problems where the Poisson’s ratio is larger than 0.25. The details are presented by Zhao et al. (2010a) and Zhao (2010). The behavior of the normal spring is in the conventional way,

0 2000 4000 6000 8000 10000 200 400 600 800 1000 crack velocity [m/s]

initial stored energy (= G

f

) [N/m]

vL vR

Fig. 1 The initial stored energy (i.e., fracture energy) versus the crack propagation velocity observed in the PMMA plates

Fig. 2 The physical model (top) and the calculation cycle (bottom) of the DLSM

which means that the normal deformation is directly cal-culated from the displacement of the two linked particles. For example, assume that there exists one bond between particles i and j, and the unit normal n nx; ny; nz

 

points form particle i to particle j. The relative displacement vector is calculated as

uij¼ uj ui ð5Þ

Then normal displacement vector and interaction force between two particles is given as

un_ij¼ u ij nn and Fnij¼ knunij ð6Þ

where knis the normal spring stiffness and n is the normal to the bonds. The multi-body shear spring between two particles is introduced through a multi-body shear displacement vector: us_ij¼ e½ bondn T_{ ½ e} bondn T    n   n ð7Þ

where e½ _bondis the bond strain state which is evaluated by the DLSM (Zhao et al.2010a). Then the shear interaction between two particles is given as

Fs_ij¼ ksusij ð8Þ

where ksis the stiffness of the shear spring. Equations6–8 provide the force update for the DLSM. For displacement update, the particle velocity is calculated as

_uðtþDt=2Þ_i ¼ _uðtDt=2Þi þ

P FðtÞ_j mp

Dt ð9Þ

where _uðtþDt=2Þ_i is the particle velocity at tþ Dt=2; _uðtDt=2Þi

the particle velocity at t Dt=2; mp the particle mass,

P

FðtÞ_j the sum of all the forces acting on the particle i including external forces, and dt the time step. Finally, the updated displacement of the particle is obtained as uðtþDtÞ_i ¼ uðtÞi þ _u

ðtþDt=2Þ

i Dt ð10Þ

where uð_itþDtÞand uð Þ_itare the displacements at t ? Dt and t, respectively. Equations9and10form the main procedure involved in the DLSM.

3.1 Cohesive Law Used in the DLSM

The constitutive law implemented in the DLSM is shown in Fig.3, where u represents the norm of u, and u* is its ultimate value beyond which the bond fails. d1is the ratio of bond deformation to its ultimate deformation at the hardening point, and d2is the same ratio but corresponding to the softening point (see Fig.3).

There are three phases involved for each single bond: linear, hardening, and softening phase. The bond consti-tutive law is written as follows:

f¼ knu for u d1u knud1þ kredud2 knud1 ð Þ u  u_d 1 ð Þ u_d 2 ud1 for d1u\u d2u kredud2 u u u_{ u}_d 2 for d2u\u u 8 > > > > > < > > > > > : ð11Þ

kredis called the initial softening stiffness and is defined in Fig.3. The fracture energy, needed to fully open a unit area of a cracked surface in the DLSM model, is

Gf ¼ 1 2aE u l  2 l ð12Þ

where l* is the mean bond length in the whole DLSM model (Zhao2010), and the dimensionless coefficient a is obtained by a¼ d21þ dð 2 d1Þ d1þ kred kn d2   þkred kn 1 d2 ð Þd2 ð13Þ

Finally, the initial softening stiffness kredis calculated in terms of the material tensile strength rmaxas

kred¼ kn 1 d2 rmax E l u ð14Þ

Using Eqs.12–14, kredand u* are calculated provided; we have d1and d2. Because the obtained kred and u* are computed from the material properties (i.e., E, rmax, Gf, etc.), which are evaluated from static fracturing, we call them rate-independent parameters. The parameters d1and d2are estimated by a simple trail and error, as explained later.

3.2 Rate-Dependent Model

To introduce rate effects into the DLSM, the initial soft-ening stiffness and the ultimate bond deformation is assumed to change instantaneously with the bond defor-mation rate _u as follows:

^ kred ¼ kred 1þ _u _uk ref  n! and u^¼ u 1þ _u _uu ref  g   ð15Þ In Eq.15, ^kred and ^u are the rate-dependent values for

the initial softening stiffness and the bond ultimate deformation. _uk_ref; _uu

ref; n, and g are model parameters

which are adjusted such that the experimental results are reproduced. Their effect on the model response is discussed in Sect.6. The validity of Eq.15is examined by Kazerani and Zhao (2010) in detail.

Expansion of Eq.11 leads to Eq.16, into which substituting ^kred and ^u for kred and u gives the

rate-dependent cohesive law:

f¼ knu for u d1u knud1þ ^ kredud2 knud1   u u_d 1 ð Þ u_d 2 ud1 for d1u\u d2u ^ kredud2 ^ u u ^ u_{ u}_d 2 for d2u\u ^u 8 > > > > > < > > > > > : ð16Þ 4 Modeling

To verify the model validity and adequacy, the laboratory tests introduced in Sect. 2 are simulated. The model geometry and mechanical properties are assumed the same as those used in the laboratory.

4.1 Material Properties

The PMMA material parameters are: density q = 1,180 kg/m3, Young’s modulus E = 3,090 MPa, Poisson ratio m = 0.35, tensile strength rten= 75 MPa, and frac-ture energy Gf= 300 N/m. According to the elastic wave theory (Bedford and Drumheller1996), longitudinal, shear, and Rayleigh surface wave speed are obtained as cL= 1,618 m/s, cS= 985 m/s, and cR= 906 m/s, respectively. Note that the experimentally obtained limiting velocity of crack propagation, vL, is about 70% of cR.

4.2 Model Micro-Parameters and Geometry

The DLSM model is fully three-dimensional. The PMMA plate modeled is l = 32 mm, h = 16 mm, and contains a 4-mm-long edge crack along the centerline (see Fig.4).

The plate is made of 512,000 particles with diameter of 0.1 mm. As seen in Fig.2, all the particles have the same size and they are regularly packed. Therefore, the entire bonds have the same length, which is equal to the particle size, i.e., l* = 0.1 mm in Eq.12.

Regarding Eq. 2, bond stiffness in tension and shear is calculated as kn= 219.67 MN/m and ks = -65.09 MN/m. In the rate-independent cohesive law (i.e., static fractur-ing), the bond strength rmax and the ultimate bond defor-mation u* are constant. rmax= 75 MPa, which is assumed equal to the PMMA tensile strength.

As seen in Fig. 3, parameters d1= 0.358 is adjusted such that the maximum bond deformation at the end of phase-I becomes rmax=kn: Parameter d2= 0.536 is

deter-mined by a trail and error process explained in Sect. 6. Using Eqs. 12–14, the ultimate bond deformation u* = 6.78 lm and a = 0.667 are calculated to fit the given material fracture energy Gf= 300 N/m.

4.3 Boundary Conditions and Fracture Numerical Mechanism

As Fig. 4illustrates, the plate is initially loaded by a pre-scribed tensile displacement D along the upper and lower boundaries in the y-direction, while the crack is not allowed to propagate yet. The specimen is not constrained in the x-direction. A static analysis is conducted to calcu-late the initial strain and stress states of the pcalcu-late with the stationary crack. Under these loading circumstances, the strain energy W (per unit area) stored in the pre-strained plate is calculated by:

W ¼1 2 E 2Dð Þ2 h ¼ 2ED2 h ð17Þ

After the static calculation, an explicit dynamic analysis is performed without changing the boundary conditions. The time step used in the dynamic solution should be shorter than the time of P-wave propagation through the lattice length as in classical explicit based methods, e.g., DEM and FEM. To satisfy that, the time step is taken as 0.01 ls which guarantees the numerical stability of the analysis.

Note that at end of the static stage, a high stress con-centration is induced at the crack tip because of the pre-existing crack. This stress far exceeds the bond strength

x y Δ Δ l h

and may lead to a sudden and extensive bond failure over the zone close to the crack tip during the first steps of the dynamic solution. However, after a short time, the crack will propagate in a steady state similar to what is observed in the laboratory (see Sect.6).

As soon as a bond breaks during the steady crack propagation, the force distribution within the entire particle system will adapt itself with the model new geometry. It means that the tensile stress concentration moves toward the bonds, which are located at the new position of the crack tip. This tensile stress stretches the bonds and accelerates the particles. During this phase, the bond stiffness ^kredand the ultimate displacement ^u are adjusted

according to Eq.15, and the bond force is continuously updated by Eq.16. This procedure clearly shows how the crack propagates through the particle assemblage and the rate-dependent law controls the crack speed.

5 Crack Branching (Bifurcation)

The crack surface appearance varies when the crack velocity changes. The general tendency is that the surface roughness becomes larger when the crack velocity increases.

As described by Zhou et al. (2005), under low pre-strain (D \ 0.10 mm), the crack steady velocity is low (vcr\ 450 m/s), and the fractured surface is smooth. In moderate values of pre-strain (D = 0.10–0.14 mm), the crack velocity is higher (vcr= 450–600 m/s), and some streaks and roughness are observed on the crack surface. At first, these streaks are not remarkable. However, when the crack velocity approaches 600 m/s, they become deeper (0.01 mm order), but the crack still runs in a single straight path (Zhou et al.2005; Fineberg2006).

Under the highest pre-strain (D = 0.16 mm), the crack velocity exceeds 600 m/s, and some small cracks start branching from the main crack. Generally, in the velocity range about 600–650 m/s, these small cracks stop further propagating and create the micro-branches seen in the laboratory. However, when vcr[ 650 m/s, one or few of these small cracks continue running as far as 1 mm or more, and crack bifurcation (branching) happens. A schematic representation of the crack bifur-cation, observed by Zhou et al. (2005), is presented in Fig.5.

A review on the fracture studies shows that the model mesh size cannot be lower than 0.1 mm order to have an efficient solution with the current computer facilities (Xu and Needleman1994,1995,1996; Zhai et al.2004; Zhou et al.2005; Block et al. 2007; Kazerani and Zhao 2010; Zhao and Zhao 2009). The streaks and micro-branches,

observed on the crack surface, are much smaller than this value for low and moderate pre-strain. Therefore, crack branching could not be allowed in the numerical model except for the highest pre-strain where the branch size is larger than the mesh resolution.

6 Calculation Results and Discussion

The experimental results presented in Fig.1are compared with the predictions using the rate-independent (RI) and rate-dependent (RD) models.

In these models, crack propagation under six different loadings is determined. The prescribed boundary dis-placement D is chosen to be 0.06, 0.08, 0.10, 0.12, 0.14, and 0.16 mm. According to Eq.17, the stored energy W is 1,391, 2,472, 3,863, 5,562, 7,571, and 9,888 N/m, respectively.

As shown in Fig.3, the softening length in phase-III will decrease with increasing d2. However, we found that when d2exceeds a threshold, equal to 0.536, the crack will arrest for the lowest pre-strain (D = 0.06 mm) in the rate-inde-pendent model. For this reason, d2 can be finally deter-mined from a trail and error process which finally led to d2= 0.536.

There are two rate-dependent (RD) cases possible: • Partial rate-dependency, RD-P: Only u* is

rate-depen-dent, while kredis held constant.

• Full rate-dependency, RD-F: Both kredand u* change in terms of the bond deformation rate.

Results using the RI case are presented in Fig.6in terms of the crack tip position versus D where the crack speed for each value of D is computed by the average slope of the corresponding curve. As seen, the crack propagates in a steady state, and no fluctuation or instability is observed in the crack speed response.

Δ = 0.16 mm

Δ = 0.16 mm

Fig. 5 A representative crack bifurcation observed in the highest pre-strain

6.1 Partial Rate-Dependent Case

The RD-P modeling logic is similar to the cohesive ele-ment model by Zhou et al. (2005) and the cohesive nodal force model of Block et al. (2007). In Fig.7, the experi-mental results are compared with those of the RI and RD-P model, with _uu_ref= 16 m/s and g = 1.0. As seen, the crack velocity in the RI model is about the Rayleigh wave speed, which is much higher than the limiting velocity vL observed in the laboratory. This proves that the rate-inde-pendent models are not applicable to dynamic fracturing, and the rate effects have to be introduced into the model.

Although the RD-P model results in better predictions, it cannot precisely reproduce the laboratory results. As argued by Zhou et al. (2005), numerically obtained crack speed directly depends on the magnitude of the fracture energy Gf released at each bond failure. Since the bond strength is held constant in the RD-P model, Gfincrease is limited to the increase in u*. Hence, the fracture energy release cannot be adequately reproduced.

Zhou et al. (2005) claimed that the crack branching allowance solves this problem. Block et al. (2007) assigned

a damage model to the solid material to increase the fracture energy and control the crack speed. These two methods artificially treat the discussed lack of energy dis-sipation through the bond failure. As described in Sect.5, Zhou’s solution (i.e., crack branching allowance) is not in agreement with the specimen post-fracture appearance in low and moderate pre-strain where no branching is observed in the plate. Application of a damage model for solid elements by Block et al. (2007) does not sound quite convincing, because the majority of the energy dissipation seems to happen within the damaged solid material, not in the cracked surface.

One may think that the RD-P model may lead to better results by changing g or _uu

ref. However, as discussed by

Zhou et al. (2005), this model never properly reproduces the laboratory results, and sometimes causes an unrealis-tically high bond deformation that even exceeds the pre-scribed boundary displacement (i.e., 2D).

6.2 Full Rate-Dependent Case

As observed in the laboratory, cracking in a fast moving crack-tip zone may result in local stress increase (Costanzo and Walton 1998; Allen and Searcy 2001; Kubair et al.

2003; Xu et al.2003; Ivankovic et al.2004). A quantitative evaluation, performed by Zhou and Molinari (2004), shows that the failure strength of ceramics increases up to 15% when the strain rate increases from 40 to 5,000 s-1. This stress increase raises Gfand consequently influences on the material fracturing behavior.

All the previous discussions make us believe that each bond break must dissipate enough energy to limit the crack speed, and both the bond cohesive strength and deforma-tion must be increased to produce the actual fracture energy. This idea led to the development of a full rate-dependent model (RD-F) in which both u* and kredchange with _u:

A comparison of the laboratory results, RD-P and RD-F simulations, is shown in Fig.8. The rate-dependency

0 4 8 12 16 20 24 0 10 20 30 time [microsec]

crack tip location [mm]

Δ: 0.06 Δ: 0.08 Δ: 0.10 Δ: 0.14 Δ: 0.16

Fig. 6 The crack tip location versus time in the rate-independent (RI) case 0 2000 4000 6000 8000 10000 12000 0 200 400 600 800 1000 crack velocity [m/s] stored energy [N/m] empirical fit RI RD-P

Fig. 7 The variation of the crack velocity versus the initial stored energy of the plate for the independent (RI) and the partial rate-dependent (RD-P) model 0 2000 4000 6000 8000 10000 12000 0 200 400 600 800 1000 crack velocity [m/s] stored energy [N/m] empirical fit RD-P RD-F

Fig. 8 Variation of the crack velocity versus the initial stored energy of the plate for the full rate-dependent (RD-F) model

parameters are taken as _uu

ref = 40.0 m/s, _ukref = 5.0 m/s,

g = 4.0, and n = 0.2 by trail and error to get the best fit. As seen in Fig.8, while the RD-P solution approaches unrealistically high cracking speed for large values of stored energy, the RD-F follows the experimental data and closely predicts the PMMA fracture response.

According to the discussion in Sect. 5, no crack branching is permitted in the model for low and moderate pre-strain, and the crack propagates in a straight path only. However, in the highest value of prescribed loading (D = 0.16 mm), the crack is allowed to bifurcate. The results of branching for this case are presented in Fig.9. It shows that the numerically obtained branching pattern is quite similar to the laboratory results observed by Shioya and Zhou (1995) and Zhou et al. (2005) (see Fig.5).

7 Conclusions

Dynamic fracture behavior of PMMA, a brittle material, was explored using a rate-dependent model implemented in the DLSM. The proposed model increases both the ultimate deformation and the strength of the bonds in terms of their deformation rate.

Since the magnitude of the micro-branches, experi-mentally observed on the cracked surface for low and moderate pre-strain, is much smaller than the finest pos-sible mesh size, no crack branching was allowed, and the model assumes the crack to propagate through a straight path within the plate. However, as the crack branch size in high pre-strain is comparable with the model mesh reso-lution, the crack bifurcation was allowed in this situation.

Fig. 9 Distribution of the y-direction strain, predicted by the DLSM at t = 5 ls (a) and t = 25 ls (b) for the highest pre-strain

The model reproduces the actual energy dissipation through the cracked surface. It also predicts the laboratory crack velocity for all values of the prescribed loading, as well as the branching pattern observed in high pre-strain. All these achievements are reached while no additional assumptions, like introducing a damage model (Block et al.

2007) or unnecessary branching (Zhou et al. 2005), are required.

The obtained results verify the validity and the adequacy of the proposed model to effectively predict the fracture behavior of the brittle materials.

Acknowledgments This research is sponsored by the fund of the Swiss National Science Foundation (SNSF) and the Ecole Polytech-nique Fe´de´rale de Lausanne (EPFL). The financial support from the China Scholarship Council to G.F Zhao is also acknowledged. The authors are grateful to the anonymous reviewer for his/her construc-tive comments and valuable time devoted to improving their manuscript.

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